| L(s) = 1 | + 27·3-s − 76.3·5-s + 439.·7-s + 729·9-s + 3.31e3·11-s + 1.15e4·13-s − 2.06e3·15-s + 1.70e4·17-s + 9.14e3·19-s + 1.18e4·21-s + 4.55e4·23-s − 7.22e4·25-s + 1.96e4·27-s − 1.85e5·29-s + 5.12e4·31-s + 8.94e4·33-s − 3.35e4·35-s + 1.59e5·37-s + 3.12e5·39-s − 5.48e4·41-s − 3.75e5·43-s − 5.56e4·45-s − 2.71e5·47-s − 6.30e5·49-s + 4.61e5·51-s + 3.62e5·53-s − 2.52e5·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.273·5-s + 0.484·7-s + 0.333·9-s + 0.750·11-s + 1.46·13-s − 0.157·15-s + 0.843·17-s + 0.305·19-s + 0.279·21-s + 0.780·23-s − 0.925·25-s + 0.192·27-s − 1.40·29-s + 0.308·31-s + 0.433·33-s − 0.132·35-s + 0.517·37-s + 0.843·39-s − 0.124·41-s − 0.720·43-s − 0.0910·45-s − 0.381·47-s − 0.765·49-s + 0.486·51-s + 0.334·53-s − 0.205·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(3.535504308\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.535504308\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 27T \) |
| good | 5 | \( 1 + 76.3T + 7.81e4T^{2} \) |
| 7 | \( 1 - 439.T + 8.23e5T^{2} \) |
| 11 | \( 1 - 3.31e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 1.15e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.70e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 9.14e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 4.55e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.85e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 5.12e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.59e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 5.48e4T + 1.94e11T^{2} \) |
| 43 | \( 1 + 3.75e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.71e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 3.62e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.68e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.65e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.02e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.63e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 1.87e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 6.03e6T + 1.92e13T^{2} \) |
| 83 | \( 1 + 2.32e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.23e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 1.18e7T + 8.07e13T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.04416500040502316888613928056, −9.128630352551216589153533602239, −8.318350793715735130418876806873, −7.54060486801615300184582613536, −6.42578287853184756710345704909, −5.33171183833560836054143137339, −3.99612152191859405636831761355, −3.34742435004788176262483216696, −1.82265125171952310098522853670, −0.913151859014131801593093573727,
0.913151859014131801593093573727, 1.82265125171952310098522853670, 3.34742435004788176262483216696, 3.99612152191859405636831761355, 5.33171183833560836054143137339, 6.42578287853184756710345704909, 7.54060486801615300184582613536, 8.318350793715735130418876806873, 9.128630352551216589153533602239, 10.04416500040502316888613928056
